Question

8. In right triangle LMN shown below, altitude MK is drawn to LN from M. IF LM = 12 and LK =10, then
which of the following is the length of KN?
(1) 2.8
(2) 4.4
(3) 6.2
(4) 14.4

Answer 1

The length of the KN is 4.4

Step-by-step explanation:

We know from Pythagoras theorem  

In a right angle ΔLMN  

Base² + perpendicular² = hypotenuse ²

From the properties of triangle we also know that altitudes are ⊥ on the sides they fall.

Hence ∠LKM = ∠NKM = 90 °

Given values-

LM=12

LK=10

Let KN be “s”

⇒LN= LK + KN

⇒LN= 10+x        eq 1

Coming to the Δ LKM

⇒LK²+MK²= LM²

⇒MK²= 12²-10²

⇒MK²= 44           eq 2

Now in Δ MKN

⇒MK²+ KN²= MN²

⇒44+s²= MN²       eq 3

In Δ LMN

⇒LM²+MN²= LN²

Using the values of MN² and LN² from the previous equations

⇒12² + 44+s²= (10+s) ²

⇒144+44+s²= 100+s²+20s

⇒188+s²= 100+s²+20s    cancelling the common term “s²”

⇒20s= 188-100

∴ s= 4.4

Hence the value of KN is 4.4  

Answer 2

         Measure of segment KN will be 4.4 units.

         Option (2) will be the correct option.

Application of Geometrical mean theorem in a right triangle:

• In a right triangle LMN given with altitude MK drawn from M to LN,

         By applying geometrical mean theorem in right triangle LMN,

         MK = $\sqrt{LK\times KN}$

Given in the question,

• In a right triangle LMN, MK is an altitude.
• Measure of LK = 10
• Measure of LM = 12

By applying Pythagoras theorem in right triangle LKM,

(LM)² = (LK)² + (MK)²

(12)² = (10)² + (MK)²

MK = √(144 - 100)

      = √44

By applying geometrical mean theorem,

MK = $\sqrt{LK\times KN}$

$\sqrt{44}=\sqrt{10\times KN}$

44 = 10 × KN

KN = 4.4

     Therefore, measure of segment KN will be 4.4 units.

                        Option (2) will be the correct option.

Learn more about the application of geometrical theorem here,

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